where \(\beta =\frac{1}{k_B T}\)  is the inverse thermal energy.
Finally, we invoke the secular approximation, dropping the terms in the Redfield tensor, \(R_{\mu \nu ,\mu ^{\prime }\nu ^{\prime }} \) , for which \(\omega _{\mu \nu } -\omega _{\mu ^{\prime }\nu ^{\prime }}\ne 0\) .
Employing the secular approximation ensures the positivity of the reduced density matrix, i.e. the diagonal matrix elements are always positive [1]}, [2]}.
In addition to preserving positivity, the secular approximation also guarantees the long-time equilibrium state is given by the Boltzmann state of the system Hamiltonian,
\(\rho (t \rightarrow \infty ) = \frac{\mbox{e}^{-\beta H^{\rm (s)}} }{\mbox{tr}[\mbox{e}^{-\beta H^{\rm (s)}}]}\) .
In this letter, we use \(\omega _c=0.004\) meV and \(\gamma =3\times 10^{-8}\) meV. The hyperfine magnetic field, \(\vec{B}^{\rm (hf)}\) , is drawn from a 3D Gaussian distribution with a standard deviation of \(1 \mathrm {mT}\) .
